{"title":"有界d维域上分数阶泊松方程的球上行走驱动有限差分方法","authors":"Daxin Nie, Jing Sun, Weihua Deng","doi":"10.1093/imanum/draf031","DOIUrl":null,"url":null,"abstract":"Inspired by the idea of ‘walk-on-sphere’ algorithm, we propose a novel finite-difference framework for solving the fractional Poisson equation under the help of the Feynman-Kac representation of its solution, i.e., walk-on-sphere-motivated finite-difference scheme. By choosing suitable basis functions in interpolatory quadrature and using graded meshes, the convergence rates can achieve up to $O(h^{2})$ in arbitrary $d$-dimensional bounded Lipschitz domain satisfying the exterior ball condition, where $d>1$; while the convergence rate can reach $O(h^{10})$ in 1-dimensional bounded domain under some regularity assumptions on the source term $f$. Furthermore, we propose a strict convergence analysis and several numerical examples in different domains, including circle, L-shape, pentagram and ball, are provided to illustrate the effectiveness of the above built scheme.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"8 1","pages":""},"PeriodicalIF":2.3000,"publicationDate":"2025-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A walk-on-sphere-motivated finite-difference method for the fractional Poisson equation on a bounded d-dimensional domain\",\"authors\":\"Daxin Nie, Jing Sun, Weihua Deng\",\"doi\":\"10.1093/imanum/draf031\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Inspired by the idea of ‘walk-on-sphere’ algorithm, we propose a novel finite-difference framework for solving the fractional Poisson equation under the help of the Feynman-Kac representation of its solution, i.e., walk-on-sphere-motivated finite-difference scheme. By choosing suitable basis functions in interpolatory quadrature and using graded meshes, the convergence rates can achieve up to $O(h^{2})$ in arbitrary $d$-dimensional bounded Lipschitz domain satisfying the exterior ball condition, where $d>1$; while the convergence rate can reach $O(h^{10})$ in 1-dimensional bounded domain under some regularity assumptions on the source term $f$. Furthermore, we propose a strict convergence analysis and several numerical examples in different domains, including circle, L-shape, pentagram and ball, are provided to illustrate the effectiveness of the above built scheme.\",\"PeriodicalId\":56295,\"journal\":{\"name\":\"IMA Journal of Numerical Analysis\",\"volume\":\"8 1\",\"pages\":\"\"},\"PeriodicalIF\":2.3000,\"publicationDate\":\"2025-05-16\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"IMA Journal of Numerical Analysis\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1093/imanum/draf031\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"IMA Journal of Numerical Analysis","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1093/imanum/draf031","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
A walk-on-sphere-motivated finite-difference method for the fractional Poisson equation on a bounded d-dimensional domain
Inspired by the idea of ‘walk-on-sphere’ algorithm, we propose a novel finite-difference framework for solving the fractional Poisson equation under the help of the Feynman-Kac representation of its solution, i.e., walk-on-sphere-motivated finite-difference scheme. By choosing suitable basis functions in interpolatory quadrature and using graded meshes, the convergence rates can achieve up to $O(h^{2})$ in arbitrary $d$-dimensional bounded Lipschitz domain satisfying the exterior ball condition, where $d>1$; while the convergence rate can reach $O(h^{10})$ in 1-dimensional bounded domain under some regularity assumptions on the source term $f$. Furthermore, we propose a strict convergence analysis and several numerical examples in different domains, including circle, L-shape, pentagram and ball, are provided to illustrate the effectiveness of the above built scheme.
期刊介绍:
The IMA Journal of Numerical Analysis (IMAJNA) publishes original contributions to all fields of numerical analysis; articles will be accepted which treat the theory, development or use of practical algorithms and interactions between these aspects. Occasional survey articles are also published.